Localization via Quasi-Periodic Bulk-Bulk Correspondence

TL;DR

This paper establishes a topological bulk-bulk correspondence linking quasi-periodic localization in the Almost Mathieu model to chiral edge modes, providing a new perspective on the metal-insulator transition distinct from Anderson localization.

Contribution

It introduces a novel topological framework connecting quasi-periodic eigenfunction localization to bulk-boundary correspondence, extending understanding of the MIT in quasi-periodic systems.

Findings

Localization linked to topological edge modes

Reduction of transfer matrix equations to rational approximations

Differentiation from Anderson localization in disordered systems

Abstract

We report on a direct connection between quasi-periodic topology and the Almost Mathieu (Andre-Aubry) metal insulator transition (MIT). By constructing quasi-periodic transfer matrix equations from the limit of rational approximate projected Green's functions, we relate results from $\text{SL}(2,\mathbb{R})$ co-cycle theory (transfer matrix eigenvalue scaling) to consequences of rational band theory. This reduction links the eigenfunction localization of the MIT to the chiral edge modes of the Hofstadter Hamiltonian, implying the localized phase roots in a topological "bulk-bulk" correspondence, a bulk-boundary correspondence between the 1D AAH system (boundary) and its 2D parent Hamiltonian (bulk). This differentiates quasi-periodic localization from Anderson localization in disordered systems. Our results are widely applicable to systems beyond this paradigmatic model.